To understand which statement is true about the inequality representing Jachin's savings, let's first set up the inequality based on the given information.
Jachin needs to save more than $400. He currently has $85 and saves $25 each week. So, after \( w \) weeks, he will have:
\[ 85 + 25w \]
We set up the inequality for the total savings to be more than $400:
\[ 85 + 25w > 400 \]
Now, let's solve for \( w \):
- Subtract 85 from both sides:
\[ 25w > 400 - 85 \]
\[ 25w > 315 \]
- Divide both sides by 25:
\[ w > \frac{315}{25} \]
\[ w > 12.6 \]
This means Jachin needs to save for more than 12.6 weeks.
Now, let's analyze the possible responses based on this result:
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"The solution w>12.6 means that it will take Jachin a minimum of 12.6 weeks to save enough money to buy the tablet." - This statement is incorrect as "minimum of 12.6 weeks" implies he can take exactly 12.6 weeks, which is not possible since \( w \) must be a whole number; he must save for more than 12.6 weeks.
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"The solution w>19.4 means that it will take Jachin a minimum of 20 weeks to save enough money to buy the tablet." - This statement is incorrect because the solution does not indicate anything above 19.4 weeks.
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"The solution w>19.4 means that it will take Jachin no more than 20 weeks to save enough money to buy the tablet." - This statement is incorrect because the solution indicates a minimum, not a maximum.
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"The solution w>12.6 means that it will take Jachin no more than 13 weeks to save enough money to buy the tablet." - This statement is incorrect for the same reason given above.
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"The solution w>19.4 means that it will take Jachin a minimum of 19.4 weeks to save enough money to buy the tablet." - Again, incorrect, because the inequality is not based on 19.4.
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"The solution w>12.6 means that it will take Jachin a minimum of 13 weeks to save enough money to buy the tablet." - This statement may seem close but it's misleading because while Jachin will indeed need 13 weeks (as he needs more than 12.6), "minimum of 13 weeks" could suggest he strictly needs 13 weeks, which is inaccurate since he could take longer.
Considering these assessments, none of the statements represent the true situation accurately. The correct interpretation is that Jachin needs more than 12.6 weeks, which rounds up to a minimum of 13 weeks, but none of the provided options correctly conveys the context of the solution.